#8095221

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(12,20)(16,21)(18,24)(22,23)(25,26,31,36)(27,28,33,32)(29,34,30,35)(37,39,38,40);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(27,33)(28,32)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,31,33)(26,32,36,28)(29,37,30,38)(34,40,35,39);(1,3)(2,6)(4,8)(5,7)(25,28,31,32)(26,33,36,27)(29,39,30,40)(34,38,35,37);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,30)(26,34,36,35)(27,37,33,38)(28,39,32,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,29)(26,35,36,34)(27,38,33,37)(28,40,32,39)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002417);